Algebraic Proof of 4-Colour Theorem? - MathOverflow

Algebraic Proof of 4-Colour Theorem?

2010-03-24T21:59:25Z

4-Colour Theorem. Every planar graph is 4-colourable.

This theorem of course has a well-known history. It was first proven by Appel and Haken in 1976, but their proof was met with skepticism because it heavily relied on the use of computers. The situation was partially remedied 20 years later, when Robertson, Sanders, Seymour, and Thomas published a new proof of the theorem. This new proof still relied on computer analysis, but to such a lower extent that their proof was actually verifiable. Finally, in 2005, Gonthier and Werner used the Coq proof assistant to formalize a proof, so I suppose only the most die hard skeptics remain.

My question stems from reading this paper by Robin Thomas. In it, he describes several interesting reformulations of the 4-colour theorem. Here is one:

Note that the cross-product on vectors in $\mathbb{R}^3$ is not an associative operation. We therefore define a bracketing of a cross-product $v_1 \times \dots v_n$ to be a set of brackets which makes the product well-defined.

Theorem. Let $i, j, k$ be the standard unit vectors in $\mathbb{R}^3$. For any two different bracketings of the product $v_1 \times \dots \times v_n$, there is an assignment of $i,j,k$ to $v_1, \dots, v_n$ such that the two products are equal and non-zero.

The surprising fact is that this innocent looking theorem implies the 4-colour theorem.

Question. Is anyone working on an algebraic proof of the 4-colour theorem (say by trying to prove the above theorem)? If so, what techniques are involved? What partial progress has been made? Or do most people consider the effort/reward ratio of such an endeavor to be too high?

I think it would be interesting to have an algebraic proof, even a very long one, particularly if the algebraic proof does not use computers. Given its connection to many other areas (Temperley-Lieb Algebras), the problem seems to be amenable to other forms of attack.

Answer by David Lehavi for Algebraic Proof of 4-Colour Theorem?

2010-03-24T22:08:18Z

Does "Lie Algebras and the Four Color Theorem" by Dror Bar-Natan qualify ?

Answer by Igor Pak for Algebraic Proof of 4-Colour Theorem?

2010-03-25T04:19:58Z

There is a classical approach by Birkhoff and Lewis, which remained dormant for decades. It was recently revived by Cautis and Jackson (start here and proceed here), using the Temperley-Lieb algebra.

Answer by Noah Snyder for Algebraic Proof of 4-Colour Theorem?

2010-03-25T17:57:53Z

I must admit I'm a bit baffled about what the question is here, and about why so many people have voted it up. What are you looking for in an answer? I don't think it's appropriate to post speculation on the internet about which mathematicians are privately working on which big problems. As to public work, you seem to have a weirdly restrictive view of what "working on" and "partial progress" mean that don't fit with my understanding of how mathematics works. Several papers have been written on the subject of possible algebraic proofs of the 4-color theorem (look at google scholar or Mathscinet for papers which cite the Saleur-Kauffman paper mentioned in the paper you're reading), but if the Bar-Natan paper doesn't count for you then you're likely to be disappointed by all of them.

The long and short of it is that everyone in quantum topology would love to prove the 4-color theorem and occasionally thinks about it. There's lots of tantalizing clues that an algebraic argument has promise, but if anyone knew how to prove it they'd have done so. As far as I know there isn't anyone who is holed up in their attic thinking about only the 4-color theorem, instead there's a lot of people who every time they find a new tool think "hrm, I wonder if this tool would work on the 4-color theorem?"

Answer by guest for Algebraic Proof of 4-Colour Theorem?

2010-04-02T22:13:16Z

An article in Scientific American, Jan 2003 offered a supposed counterexample, that sparked my interest in the problem. That and the complexity of the Appel and Haken proof motivated me to do my own study. It has minimal math, but is consistent, and approaches the problem from an entirely different direction. If interested it's at insight.awardspace.info.

Answer by Anatoly for Algebraic Proof of 4-Colour Theorem?

2012-10-20T16:16:19Z

Algebraic Proof of 4-Colour Theorem? see please http://www.math.accent.kiev.ua/article/00/4ct-2-.htm